Recent advances in the probabilistic analysis of graph-theoretic algorithms

Abstract

This talk is a survey of a research area at the interface between concrete complexity theory and the theory of random graphs. It concerns the construction of graphtheoretic algorithms which are efficient on the average, when presented with inputs drawn from a well-defined probability distribution. We present such algorithms for the following problems: I) putting a graph in canonical form with respect to isomorphism; 2) computing a maximum flow in a network; 3) finding a Hamiltonian circuit in a graph or digraph; 4) computing the connected and biconnected components of a graph; 5) computing the strongly connected components and the reachability relation of a digraph; 6) computing a perfect matching of minimum weight in a bipartite graph; 7) obtaining a good approximate solution to the directed traveling-salesman problem. We then observe that no fast-average-time algorithms are known for constructing minimum colorings or maximum cliques, and we discuss some of the impediments to constructing such algorithms.